| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | lsmelvalm.t | . . 3
⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | 
| 2 |  | lsmelvalm.u | . . 3
⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | 
| 3 |  | eqid 2737 | . . . 4
⊢
(+g‘𝐺) = (+g‘𝐺) | 
| 4 |  | lsmelvalm.p | . . . 4
⊢  ⊕ =
(LSSum‘𝐺) | 
| 5 | 3, 4 | lsmelval 19667 | . . 3
⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑥 ∈ 𝑈 𝑋 = (𝑦(+g‘𝐺)𝑥))) | 
| 6 | 1, 2, 5 | syl2anc 584 | . 2
⊢ (𝜑 → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑥 ∈ 𝑈 𝑋 = (𝑦(+g‘𝐺)𝑥))) | 
| 7 | 2 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑈 ∈ (SubGrp‘𝐺)) | 
| 8 |  | eqid 2737 | . . . . . . . . 9
⊢
(invg‘𝐺) = (invg‘𝐺) | 
| 9 | 8 | subginvcl 19153 | . . . . . . . 8
⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑈) → ((invg‘𝐺)‘𝑥) ∈ 𝑈) | 
| 10 | 7, 9 | sylan 580 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑥 ∈ 𝑈) → ((invg‘𝐺)‘𝑥) ∈ 𝑈) | 
| 11 |  | eqid 2737 | . . . . . . . . 9
⊢
(Base‘𝐺) =
(Base‘𝐺) | 
| 12 |  | lsmelvalm.m | . . . . . . . . 9
⊢  − =
(-g‘𝐺) | 
| 13 |  | subgrcl 19149 | . . . . . . . . . . 11
⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | 
| 14 | 1, 13 | syl 17 | . . . . . . . . . 10
⊢ (𝜑 → 𝐺 ∈ Grp) | 
| 15 | 14 | ad2antrr 726 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑥 ∈ 𝑈) → 𝐺 ∈ Grp) | 
| 16 | 11 | subgss 19145 | . . . . . . . . . . . 12
⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺)) | 
| 17 | 1, 16 | syl 17 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐺)) | 
| 18 | 17 | sselda 3983 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ (Base‘𝐺)) | 
| 19 | 18 | adantr 480 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑥 ∈ 𝑈) → 𝑦 ∈ (Base‘𝐺)) | 
| 20 | 11 | subgss 19145 | . . . . . . . . . . 11
⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺)) | 
| 21 | 7, 20 | syl 17 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑈 ⊆ (Base‘𝐺)) | 
| 22 | 21 | sselda 3983 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝐺)) | 
| 23 | 11, 3, 12, 8, 15, 19, 22 | grpsubinv 19030 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑥 ∈ 𝑈) → (𝑦 −
((invg‘𝐺)‘𝑥)) = (𝑦(+g‘𝐺)𝑥)) | 
| 24 | 23 | eqcomd 2743 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑥 ∈ 𝑈) → (𝑦(+g‘𝐺)𝑥) = (𝑦 −
((invg‘𝐺)‘𝑥))) | 
| 25 |  | oveq2 7439 | . . . . . . . 8
⊢ (𝑧 = ((invg‘𝐺)‘𝑥) → (𝑦 − 𝑧) = (𝑦 −
((invg‘𝐺)‘𝑥))) | 
| 26 | 25 | rspceeqv 3645 | . . . . . . 7
⊢
((((invg‘𝐺)‘𝑥) ∈ 𝑈 ∧ (𝑦(+g‘𝐺)𝑥) = (𝑦 −
((invg‘𝐺)‘𝑥))) → ∃𝑧 ∈ 𝑈 (𝑦(+g‘𝐺)𝑥) = (𝑦 − 𝑧)) | 
| 27 | 10, 24, 26 | syl2anc 584 | . . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑥 ∈ 𝑈) → ∃𝑧 ∈ 𝑈 (𝑦(+g‘𝐺)𝑥) = (𝑦 − 𝑧)) | 
| 28 |  | eqeq1 2741 | . . . . . . 7
⊢ (𝑋 = (𝑦(+g‘𝐺)𝑥) → (𝑋 = (𝑦 − 𝑧) ↔ (𝑦(+g‘𝐺)𝑥) = (𝑦 − 𝑧))) | 
| 29 | 28 | rexbidv 3179 | . . . . . 6
⊢ (𝑋 = (𝑦(+g‘𝐺)𝑥) → (∃𝑧 ∈ 𝑈 𝑋 = (𝑦 − 𝑧) ↔ ∃𝑧 ∈ 𝑈 (𝑦(+g‘𝐺)𝑥) = (𝑦 − 𝑧))) | 
| 30 | 27, 29 | syl5ibrcom 247 | . . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑥 ∈ 𝑈) → (𝑋 = (𝑦(+g‘𝐺)𝑥) → ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 − 𝑧))) | 
| 31 | 30 | rexlimdva 3155 | . . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → (∃𝑥 ∈ 𝑈 𝑋 = (𝑦(+g‘𝐺)𝑥) → ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 − 𝑧))) | 
| 32 | 8 | subginvcl 19153 | . . . . . . . 8
⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑧 ∈ 𝑈) → ((invg‘𝐺)‘𝑧) ∈ 𝑈) | 
| 33 | 7, 32 | sylan 580 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑧 ∈ 𝑈) → ((invg‘𝐺)‘𝑧) ∈ 𝑈) | 
| 34 | 18 | adantr 480 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑧 ∈ 𝑈) → 𝑦 ∈ (Base‘𝐺)) | 
| 35 | 21 | sselda 3983 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ (Base‘𝐺)) | 
| 36 | 11, 3, 8, 12 | grpsubval 19003 | . . . . . . . 8
⊢ ((𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑦 − 𝑧) = (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧))) | 
| 37 | 34, 35, 36 | syl2anc 584 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑧 ∈ 𝑈) → (𝑦 − 𝑧) = (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧))) | 
| 38 |  | oveq2 7439 | . . . . . . . 8
⊢ (𝑥 = ((invg‘𝐺)‘𝑧) → (𝑦(+g‘𝐺)𝑥) = (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧))) | 
| 39 | 38 | rspceeqv 3645 | . . . . . . 7
⊢
((((invg‘𝐺)‘𝑧) ∈ 𝑈 ∧ (𝑦 − 𝑧) = (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧))) → ∃𝑥 ∈ 𝑈 (𝑦 − 𝑧) = (𝑦(+g‘𝐺)𝑥)) | 
| 40 | 33, 37, 39 | syl2anc 584 | . . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑧 ∈ 𝑈) → ∃𝑥 ∈ 𝑈 (𝑦 − 𝑧) = (𝑦(+g‘𝐺)𝑥)) | 
| 41 |  | eqeq1 2741 | . . . . . . 7
⊢ (𝑋 = (𝑦 − 𝑧) → (𝑋 = (𝑦(+g‘𝐺)𝑥) ↔ (𝑦 − 𝑧) = (𝑦(+g‘𝐺)𝑥))) | 
| 42 | 41 | rexbidv 3179 | . . . . . 6
⊢ (𝑋 = (𝑦 − 𝑧) → (∃𝑥 ∈ 𝑈 𝑋 = (𝑦(+g‘𝐺)𝑥) ↔ ∃𝑥 ∈ 𝑈 (𝑦 − 𝑧) = (𝑦(+g‘𝐺)𝑥))) | 
| 43 | 40, 42 | syl5ibrcom 247 | . . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑧 ∈ 𝑈) → (𝑋 = (𝑦 − 𝑧) → ∃𝑥 ∈ 𝑈 𝑋 = (𝑦(+g‘𝐺)𝑥))) | 
| 44 | 43 | rexlimdva 3155 | . . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → (∃𝑧 ∈ 𝑈 𝑋 = (𝑦 − 𝑧) → ∃𝑥 ∈ 𝑈 𝑋 = (𝑦(+g‘𝐺)𝑥))) | 
| 45 | 31, 44 | impbid 212 | . . 3
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → (∃𝑥 ∈ 𝑈 𝑋 = (𝑦(+g‘𝐺)𝑥) ↔ ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 − 𝑧))) | 
| 46 | 45 | rexbidva 3177 | . 2
⊢ (𝜑 → (∃𝑦 ∈ 𝑇 ∃𝑥 ∈ 𝑈 𝑋 = (𝑦(+g‘𝐺)𝑥) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 − 𝑧))) | 
| 47 | 6, 46 | bitrd 279 | 1
⊢ (𝜑 → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 − 𝑧))) |